van Acker, Nadine

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The maximal function of a complex measure

Institute of Mathematics AS CR, v. v. i.

WSGP 10Nadine AckerThe maximal function of a complex measureIn: Jarolím Bureš and Vladimír Souček (eds.): Proceedings of the Winter School "Geometry andPhysics". Circolo Matematico di Palermo, Palermo, 1991. Rendiconti del Circolo Matematico diPalermo, Serie II, Supplemento No. 26. pp. [41]--46.Persistent URL: http://dml.cz/dmlcz/701478Terms of use:© Circolo Matematico di Palermo, 1991Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czTHE MAXIMAL FUNCTION OF A COMPLEX MEASURE Nadine Van Acker ABSTRACT. The Marcinkiewicz Interpolation Theorem is proved in the setting of the unit sphere in euclidean space of arbitrary dimension. This leads to a key result in the study of the maximal function of a complex measure and, in particular, of an integrable function on this unit sphere. 1. INTRODUCTION In the theory of the H spaces of monogenic functions [2], which is developed in the framework of Clifford analysis [1], an important role is played by the results concerning the boundary behaviour of Poisson integrals of complex measures and, in particular, of integrable functions on the unit sphere. o JTJ Let P denote the Poisson kernel in B , the unit ball in IR ; it is given by m l-r2 P(x,y) = I I 1 |x-y| where r = |x| and y is on the unit spere S Definition 1.1. If ii is a complex measure on S , then its Poisson integral T[^i] is defined as P I M H X ) = S P(x,y) du(y) , x € Bm . sm-i In the special case where the measure [i is derived from an L -function f, this definition rewrites as . P [ f ] ( x ) = S P(x,y) f(y) dcr(y) , x € &m , cr being the normalized Lebesgue measure on S , <r(S ) = 1. The main result about the boundary behaviour of Poisson integrals is the 42 NADINE VAN ACKER so called Koranyi theorem, relating the maximum value of the continuous function T[\i] in a cone D , to the value of the maximal function of \i at the top of the cone. Let us first introduce these notions explicitly before stating the theorem. Definition 1.2. Let C € S and c > 1. The conical region D with top £ is defined by Da(?) = {y € Rm : |?-y| < ca(l - |y|)> Definition 1.3. If ji is a complex measure on S , then its maximal function Mfi : Sm-1-> [0,+oo] is given by : lnl(bk(S,0)) M ^ > " ~ P <r(bkO~,*)) where |u| is the total variation of n and b k ( £ , 0 V is a sphere segment centered at § € S with half a solid angle 0. Theorem 1.4. (Koranyi) Given a complex measure fi on S , for each conical region D there exists a constant A such that on S : a sup |̂ [fx] (x) | -S A„ Mfx(?) . x€Da(?) a Proof : see [ 2 V The aim is to study, in the third section, this maximal function of a complex measure, especially when that measure is derived from an integrable function on S . The key result on the maximal function is strongly related to the so called Marcinkiewicz Interpolation Theorem, which we treat of first in the second section. 2. THE MARCINKIEWICZ INTERPOLATION THEOREM. Let Z denote a positive measure on Sm-1. We consider an operator T, acting ~ m—i on functions f € L^S ), and mapping them into Z-measurable functions Tf : S -> [0,+ooV Moreover we assume T to be subadditive : T(f + i) -s Tf + Ti for all f,i € L^s"1"1) . Next we define numbers cr> 1 -s r -s oo f as to be the smallest constants for which the estimates THE MAXIMAL FUNCTION OF A COMPLEX MEASURE 4 3 Z({Tf > t } ) * c t " r S If |FdZ and IITfll < c llfll r 00 00 00 s m - i hold over the whole of S and for all t, 0 < t < oo. Notice that the constants c might be oo. Finally we introduce the notation K(a,b,c,...) for a constant K, depending upon the parameters a,b,c,..., and which is finite whenever all the parameters are finite. Theorem 2.1. For all f € L (Sm_1) and 1< p < r -S oo , S (Tf)P dZ -* K (ci,cr) S If lP dZ . sm-i P ' r sm-i Proof : Let F : S -> [0,+oo] be a Z-measurable function. It is then easily seen that 00 J" F(T))P dZ(n) = S ptP_1 E({F(T>) >t})dt . sm-i 0 Given t > 0, consider the decomposition of f € L (Sm_1) : f = gt + ht where £t(£) = ( 0 if |f(£)| < t „ „ 1 f(€) if l?(?)l * t and where h. = f - g.. We define G(t) = Z({Tit > ̂ }) and H(t) = Z({Tht > ^}). 2 C1 Let G (t) denote —— S If IdZ , and, for given r < oo , 1 r |f|-*t 2c H(t) = — X S IfTdZ . 1 tr |f|<t It is then clear that -> 2ci G(t) a cf * S Igtld2 s ---• J If IdS = Gt(t) , sm-l |f|2-t and similarly, H(t) -s H (t). Now applying Fubini's Theorem on 00 _ 00 S ptP *G (t)dt = 2c p S S tP"2|f |dZ dt 0 0 |f|^t oo 2c . p we find that S ptp G (t)dt = — J - J If |PdZ . 0 P sm-i 44 NADINE VAN ACKER Similarly, one gets 00 00 S ptp_1H(t)dt = 2rc p S |f(?)|rdS(?) S t p _ r _ 1 dt 0 r sm-i |f(j;)| 2rc p = — S If lPdZ r-p sm-i since p < r by assumption. In view of Tf -£ Tg + Th , it follows that Z({Tf > t}) -s S({Tit > £}) + Z({Tht > ̂ }) * Gx(t) + H^t) . Combining the obtained results we find that 00 00 00 S (Tf )pd2 = J1 ptp-1 Z({Tf > t})dt £ S ptP_1 G (t)dt + J- ptp_1 H (t)dt sm-i 0 0 0 1 s K (c ,c ) S IflPdS sm-i whence the desired result for r < +oo. If r = +oo , we may assume without loss of generality that c -s — (if r- 00 2 not, consider ~ — ) . Zc 00 It is then easily seen that llh.ll = sup | h . ( ? ) | = sup | f (?) | < t t M ^ - i * I f K t and hence IITh. II -S c II h . II is £ . t oo oo t oo 2 As Tht -s ~ a.e. on Sm~\ it follows that H(t) = Z({Tht > ̂ }) = 0 leading to S({Tf > t}) = G(t) -s G (t). Combining the obtained results, the desired result indeed follows since 00 S (Tf)PdZ = S ptP"1S({Tf > t})dt sm-i S ptP"a G (t)dt 0 2pc - J" |f lPd2 p-1 sm-i K (c ,c ) S |f|PdS p 1 00 sm-i THE MAXIMAL FUNCTION OF A COMPLEX MEASURE 45 3. THE MAXIMAL FUNCTION OF A COMPLEX MEASURE. The notion of the maximal function of a complex measure \i on S was already introduced in the first section (definition 1.3). The following two propositions may be proved along classical lines. Proposition 3.1. The maximal function MJI of the complex measure fionS is lower semi-continuous. Proposition 3.2. There exists a constant c such that for all complex measures \i on S and a l l A > 0 : „ „ II /Li II <r({Mfi > A} ) <• c — . 1 A Now consider the special case where the complex measure \i is derived from an L -function : [i = fdcr , f € L (S ) . Its maximal function also reads Mf (?) = sup — S |f(7j)|d<r(7)) 0>O G>(C,0) bk(£,tf) where w(?,0) = <r(bk(£,0)). From proposition 3.2 it now follows readily that Corollary 3.3. For a l l f € L (s"1"1) and a l l A > 0 : <r({Mf > A}) <. ^ llfll . 1 A 1 The following proposition is also easily verified : Proposition 3.4 The operator M : f 6 L (S ) h-> Mf is subadditive and moreover satisfies Mf (?) -< llfll , ? € s"1"1 . 00 Combining the above results with the Marcinkiewicz Interpolation Theorem of the second section, we finally arrive at the main result Theorem 3.5. *" m—l Let f € L ( S ) , l < p < o o , then there exists a constant A(p) such that S IMf |Pdcr < A(p) S If lPd<r . sm-i sm-i 46 NADINE VAN ACKER REFERENCES [1] BRACKX F., DELANGHE R., SOMMEN F. "Clifford Analysis", Pitman, London, 1982. [2] VAN ACKER N. "Hp Spaces of monogenic functions", Proceedings of the Second Workshop on Clifford Algebras and their Applications in Mathematical Physics, Montpellier, 1989. NADINE VAN ACKER UNIVERSITY OF GHENT SEMINAR OF MATHEMATICAL ANALYSIS SINT PIETERSNIEUWSTRAAT 39 B-9000 GENT BELGIUM 

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The maximal function of a complex measure

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Institute of Mathematics AS CR, v. v. i.